Novel approach to investigate $\eta$ decays via ${\eta }^{\mathrm{\prime }}\to \pi \pi \eta$

To avoid the impact from the background events directly from $e^{+}{e}^{-}$ annihilations or $J\mathrm{/}\psi$ decays, we propose a novel approach to investigate $\eta$ decays, in particular for its rare or forbidden decays, by using ${\eta }^{\mathrm{\prime }}\to \pi \pi \eta$ produced in $J\mathrm{/}\psi$ decays at the $\tau -$charm factories. Based on the MC studies of a few typical decays, $\eta \to \pi \pi$, $\gamma l^{+}{l}^{-}(l=e,\mu )$, $l^{+}{l}^{-}$, as well as $l^{+}{l}^{-}{\pi }^0$, the sensitivities could be obviously improved by taking advantage of the extra constraint of ${\eta }^{\mathrm{\prime }}$. Using one trillion $J\mathrm{/}\psi$ events accumulated at the Super $\tau$-Charm facility, the precision on the investigation of $\eta$ decays could be improved significantly and the observation of the rare decay $\eta \to e^{+}{e}^{-}$ is even accessable.